• Automorphisms and the fundamental operators associated with the symmetrized tridisc

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      https://www.ias.ac.in/article/fulltext/pmsc/131/0008

    • Keywords

       

      Symmetrized polydisc; automorphisms; $\tau_n$-contraction; fundamental operator tuple.

    • Abstract

       

      The automorphisms of the symmetrized polydisc $\mathbb{G}_n$ are well-known and are given in the coordinates of the polydisc in Edigarian and Zwonek (Arch. Math. 84 (2005) 364–374). We find an explicit formula for the automorphisms of Gn in its own coordinates. If $\tau$ is an automorphism of $\mathbb{G}_n$, then $\tau (S_1,\ldots , S_{n−1}, P)$ is a $\tau_n$-contraction, where a $\tau_n$-contraction is a commuting $n$-tuple of Hilbert space operators for which the closed symmetrized polydisc $\tau_n$ is a spectral set. Corresponding to every $\tau_n$-contraction $(S_1,\ldots , S_{n−1}, P)$, there exist $n − 1$ unique operators $A_1,\ldots , A_{n−1}$ such that $$S_i − S^∗_{n−i} P = D_P A_i D_P,\quad D_P = (I − P^∗P)^{1/2},$$ for $i = 1,\ldots , n − 1$. This unique $(n − 1)$-tuple $(A_1,\ldots , A_{n−1})$, which is called the fundamental operator tuple or $\mathcal{F}_O$-tuple of $(S_1, \ldots , S_{n−1}, P)$ in the literature, plays central role in every section of operator theory on $\tau_n$. We find an explicit form of the $\mathcal{F}_O$ tuple of $\tau (S_1,\ldots , S_{n−1}, P)$ when $n = 3$. We show by an example that a $\tau_n$-contraction may not have commuting $\mathcal{F}_O$-tuple. Also, we obtain a necessary and sufficient condition under which two $\tau_n$-contractions are unitarily equivalent.

    • Author Affiliations

       

      BAPPA BISAI1 SOURAV PAL1

      1. Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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